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THE SURVEYOR'S COMPANION. 



A 



anual fax Jjraditd S>nxfopxs, 



CONTAINING 



METHODS INDISPENSABLY NECESSARY 



ACTUAL FIELD OPERATIONS, 



BY E. W. BEANS, 

Norrisiown, M™*"«mery Co., Pa. 




PHILADELPHIA: 



PUBLISHED BY J. W. MOORE, 

No. 195 Chestnut Street. 
1854. 



Entered according to the Act of Congress, in the year 1854, by 

JOHN W. MOORE, 

in the Office of the Clerk of the District Court of the Eastern 

District of Pennsylvania. 



7 A 5$t 
• B3& 



Printed by Isaac Ashmead 




PKEFACE 



The want of a work on Practical Survey- 
ing has been long felt, and is generally ac- 
knowledged. The numerous publications on 
surveying would seem to preclude the neces- 
sity of anything new. But upon examina- 
tion we find the wants of the student have 
been consulted, rather than those of the prac- 
tical man. Indeed, many of the publications 
in general use appear to have been written 
by those who were engaged in the instruction 
of youth, and who were unacquainted with 
the practical part of surveying, excepting 
perhaps so far as may have been requisite 
for the information of the classes under their 
direction. 

I have conversed with many persons who 



VI PREFACE. 

have been extensively engaged in land sur- 
veying, and I have not in a single instance 
met with any one who has not expressed his 
unqualified conviction of the want of a work 
adapted to the purposes of the practical 
surveyor. Within a few years there have 
been published several works on surveying : 
amongst them may be mentioned that by 
John Gummere, a treatise which cannot be 
too strongly recommended to those who wish 
to become familiar with this subject; and 
had it been as well adapted to the wants of 
the practitioner, as to those of the student, 
no other need have been desired. Professor 
Davies' " Elements of Surveying" is an ex- 
cellent performance. Flint's " Surveying" 
also contains much useful practical informa- 
tion. But still there seems to be wanted a 
more minute detail of expedients employed 
in the field. 

With a view to supply this defect, the 



PREFACE. VU 

following pages have been written, designed 
as a suitable treatise to be placed in the 
hands of those who wish to become familiar 
with the practice of surveying. A syste- 
matic arrangement has not been followed; 
but as my object is to supply the wants of 
the practical man, (those of the student hav- 
ing already been supplied by the authors 
mentioned,) this will be a matter of minor 
importance. 

In submitting the work to the public, it is 
not pretended to be complete in itself, but 
only introductory to a subject of much im- 
portance hitherto almost entirely neglected. 
If it prove a useful auxiliary to those who 
are about assuming the responsible duties of 
practical surveyors, the object of the publica- 
tion will have been accomplished. 

E. W. BEANS. 

Norristown, Montgomery county, 
Pa., June 1853. 



A MANUAL FOR SURVEYORS. 



CHOICE OF INSTRUMENTS. 

It is of the first importance to every young 
person, about commencing the practice of Survey- 
ing, to furnish himself with suitable instruments. 
These should be of the very best character, both as 
regards the workmanship and their adaptation to 
purposes in which they are intended to be em- 
ployed. His success may depend upon this choice, 
his pretensions to accuracy, in what he may under- 
take, must rest very much upon it. Unless he is 
acquainted with the use of the Instruments in the 
field, it would be well to advise with some person 
in whose judgment he can confide. It would also 
be advisable to visit the shops of the different in- 
strument makers, and examine the various instru- 
ments in use. He may be enabled by these means 
2 



10 A MANUAL FOR SURVEYORS. 

to make a more suitable choice than he would 
otherwise do. Having the instruments before 
him, and their peculiar advantages and uses ex- 
plained, he would be enabled to select those best 
suited to his purpose. He should be provided with 
an instrument for measuring angles, and taking 
bearings, a chain, measuring-tape, plummet, read- 
ing-glass, &c. If he intends laying off town lots, he 
should have a leveling instrument and rod, also a 
20 feet frame for measuring distances accurately. 

The two pole chain is commonly used. It 
should be made of heavy wire, the links con- 
nected by double rings to prevent the chain kink- 
ing, it should be firm in all its parts, and of the 
standard length. 

The compass or circumferenter is in general use 
amongst surveyors in this country. In select- 
ing a compass, particular attention should be paid 
to the needle. If it vibrates long and settles with 
a gradual diminution in the arc of vibration, it 
moves with freedom on the centre pin ; but if it 
settles after a few vibrations or suddenly at any 
point, it does not move with sufficient freedom on 



A MANUAL FOR SURVEYORS. 11 

its centre, or is defective, and should not be used on 
valuable lands. 

To examine the divisions of the limb of the in- 
strument, set the needle to any degree ; if both 
ends coincide with the corresponding divisions on 
the opposite limbs of the compass, it shows the 
adjustment is correct on that course. In the same 
manner examine various divisions round the com- 
pass box, the coincidence of the opposite ends of 
the needle, with the same degree, is a proof of its 
being correct. If it be a nonius compass, the 
nonius may be moved to the right or left a few 
degrees ; if the needle move over the same number 
of degrees, it is a proof of the correctness of the 
graduations. The instrument may be further 
tested by laying off, with it, on the ground, a 
square, the side of which may be 50 or more 
perches. Then if the diagonals be measured and 
found equal (or very nearly so, as errors may be 
in the measured distances,) it is evidence of the 
correctness of the divisions, needle, &c, and that 
the instrument is a good one. Any regular figure 
may be used for this purpose. A nonius compass 



12 A MANUAL FOR SURVEYORS. 

is preferable to a plain one in several particulars — 
some of which may be mentioned. Having found 
the difference of variation between the present 
time, and that at which a survey was formerly 
made, set the nonius to this difference, the needle 
will show, on the face of the instrument, when the 
sights are set to the lines of survey, the same course 
as formerly, unless the compasses, by which the 
bearings were taken, differ in other particulars. 
Another reason for preferring the nonius compass 
is, that we may run a line to minutes, by setting 
the needle to the whole degrees, and the nonius to 
the minutes of the course. 

Angles of elevation, when a level is attached to 
the compass, (which it should always have,) may 
be taken, by having one of the sights graduated 
into degrees. This may be easily done by the 
surveyor himself calculating the tangents of the 
angles, to a radius, equal to the distance of the 
sights apart, and setting off those distances, on the 
sight. This may be carried to about 20°. The 
links to be deducted, for every chain length oblique 
measure, to reduce it to horizontal measure, may 



A MANUAL FOR SURVEYORS. 13 

also be marked on the sights, corresponding with 
the degrees of elevation. A compass so fitted up 
is very useful in hilly grounds ; for having an ob- 
ject at the top of the hill, we may easily take the 
angle of elevation of the hill, and thence reduce 
the oblique line measured, to the horizontal line. 

Horizontal angles may be measured with the 
nonius compass. For this purpose, procure an 
extra pair of sights, made so as to be attached to 
the face or movable part of the instrument con- 
taining the needle, and at right angles with the 
fixed sights. Bring the fixed sights to bear on an 
object in one of the lines, including the angle to be 
measured, and clamp them in that position. Un- 
clamp the nonius plate, to which the attached sights 
are affixed, and bring these sights to bear on an 
object in the other line, including the required an- 
gle. The number of degrees over which the ver- 
nier plate has moved, gives the difference between 
the angle measured and a right angle, and must be 
added to or subtracted from a right angle, accord- 
ing as the movable sights move from, or towards 
the fixed ones. 

2* 



14 A MANUAL FOR SURVEYORS. 

An arm may be also temporarily attached to the 
ball or spindle of the staff head, extending to the 
edge of the compass box, which should be divided 
up to 90°. This arm being clamped at 0° by 
means of a thumb screw, connecting the arm to the 
spindle or head of the staff, will point out the num- 
ber of degrees the compass box may move over, as 
the motion of the compass is about the immovable 
spindle of the staff head or axis of the instrument. 
A nonius compass fitted up as here directed, will 
be nearly as efficient as a theodolite, in the ordi- 
nary cases of farm surveying; and especially 
where the needle may be rendered almost useless 
from local attraction, or other accidental occur- 
rences. 

A nonius compass may be used where there is 
local attraction, as follows : Bring the sights to 
bear on an object at the back station, then move 
the nonius plate until the needle settles at the same 
course as at the last station, and clamp the nonius 
plate ; a course may then be taken in any direc- 
tion with the needle, as correctly as if free from 
the effects of local attraction. 



A MANUAL FOR SURVEYORS. 15 

The following description of an instrument for 
surveying was received from Enoch Lewis, a prac- 
tical surveyor, and author of several mathematical 
works. 

"A circumferenter for surveying in the usual 
way, and for taking horizontal angles, is formed 
with two circular plates, the lower of which is firmly 
attached to the stem of the instrument, and may 
therefore be clamped to the tripod which supports 
it ; the upper plate is connected with the sights, 
and moves with them. These circles are concen- 
tric, and are firmly attached by a screw, or moved 
the one on the other by a rack wheel. The cir- 
cumference of the lower circle is graduated, and 
the upper one contains a nonius. 

" To take an angle, place the middle of the nonius 
in conjunction with the zero of the lower plate, and 
fix them together by the screw ; then direct the 
sights along one of the lines which include the 
angle in question, and clamp the lower plate to the 
tripod. Loose the plates from each other, and 
applying the eye to the sights, turn them by means 
of the rack wheel till they coincide with the other 



16 A MANUAL FOR SURVEYORS. 

line, including the angle. The angle may then be 
read on the fixed plate by means of the nonius. 

"An instrument constructed in this manner is 
very convenient in laying out the curves of rail- 
roads. 

" A small telescopic tube fixed on the side of the 
instrument parallel to the line of sights, and move- 
able on the centre of a graduated vertical circle, 
will serve to take angles of elevation or depression 
with great facility. 

" A small table of versed sines to a radius one, 
or unity, extending to fifteen or twenty degrees, 
inserted on a leaf of the note-book, may answer 
the purpose of reduction from oblique to horizontal 
measure." 

It is generally recommended, when the compass 
is not in use, to place it in its box, where it will not 
be disturbed, and let the needle settle, and remain 
on its centre pin. However, I believe it preferable 
to let the needle settle, and then carefully screw it 
up or off the centre pin ; it will then be very nearly 
in the magnetic meridian, and will have all the 
advantages of its remaining on the centre pin, 



A MANUAL FOR SURVEYORS. 17 

without the danger of blunting it by continual fric- 
tion on the point. 

The cross is a very useful instrument to lay off 
perpendiculars when running lines or measuring 
irregular pieces of ground. A block of hard- 
grained wood, three or four inches square, and one 
and a half or two inches thick, having two saw 
kerfs cut more than half through its thickness, and 
intersecting each other at right angles at the centre 
of the block, will be sufficiently exact for the pur- 
poses above-mentioned ; which will also be of con- 
venient size to carry in the pocket. 

To use this cross, we have only to lay it on the 
face of the compass adjusted for observation, and 
directing one of the kerfs to an object; a stake set 
in the direction of the other kerf will be at right 
angles with a line joining the cross, and the object 
observed. 

The Graphometer, or Semicircle, is also a very 
useful instrument in surveying. It consists of a 
graduated semicircle, with a pair of sights in the 
direction of the diameter of the instrument, and 
having a pair of movable sights moving on the 



18 A MANUAL FOR SURVEYORS. 

centre of the semicircle, to which the verniers are 
attached. Underneath the centre of the instru- 
ment is a ball and socket to attach it to the stand 
when in use. 

To measure an angle, turn the fixed sights until 
you see an object in the direction of one of the 
lines between which the angle is to be measured, 
then turn the movable sights until an object is 
seen in the direction of the other line, the vernier 
will point out the angle between them. The socket 
has a notch in one side, so as to enable the suveyor 
to take vertical angles or altitudes. It is a very 
useful instrument for many purposes in surveying. 

In surveying valuable lands, or when great accu- 
racy is desired, a theodolite or transit should be 
used. A description of the former is deemed un- 
necessary, as this has been given in some of our 
best treatises on surveying, and also in " Simms' 
Treatise on the Principal Instruments employed in 
Surveying, Leveling, and Astronomy," a work 
which every practical mathematician should pos- 
sess. 

The " Transit" may not be so generally known. 



A MANUAL FOR SURVEYORS. 19 

It consists of two parallel plates attached to each 
other in a manner somewhat similar to the circular 
plates of a theodolite. To the upper of these cir- 
cular plates is attached a compass-box of much 
larger dimensions than that usually attached to 
the theodolite, which enables the observer to read 
the bearings from the needle much more correctly. 
The circular plates are not graduated at their 
outer edges, but so as to be read within and at the 
bottom of the compass-box. To the upper plate is 
attached the wyes, or vees, which support the axis 
of the telescope, that revolves in the same manner 
as the astronomical transit. A vertical arc is 
sometimes added to measure vertical angles. The 
instrument is furnished with clamps, tangents, 
screws, and tripod, similar to the theodolite. It 
may be used as a circumferenter in taking bearings, 
or as a theodolite for measuring angles. The ad- 
justments are few, simple, and readily made. It 
is usually provided with only one vernier; two, 
however, would be conducive to accuracy, as a 
mean of the readings would correct the eccentricity 
of the instrument. The transit may be considered, 



20 A MANUAL FOR SURVEYORS. 

for the general purposes of surveying, superior to 
any other instrument in use. 

There is one property of the magnetic needle, 
(when not disturbed by local attraction, &c.,) which 
should not be lost sight of, and that is, that any 
error committed in running one line is not commu- 
nicated to another. But when angles are taken, 
the errors may affect or be communicated to others, 
even at a remote distance from the line on which 
the error is committed. Therefore the accuracy of 
the angles must be carefully ascertained by compa- 
rison with the courses shown by the needle, other- 
wise great and perplexing errors may be intro- 
duced. 

The surveyor should be furnished with tables of 
sines, as far as 20° extending to four decimals at 
least ; nat. tangents as far as 10°, and such others 
as may be needful for reference in the field. These, 
with the Traverse Table, will be requisite to facili- 
tate many calculations necessary to be made in the 
field. Scribner's Engineer's Pocket Table Book 
may be particularly recommended. He should 



A MANUAL FOR SURVEYORS. 21 

also have a reading-glass of considerable power, to 
aid him in reading off the bearings, angles, &c, 
with precision. 



LAND MAKKS. 



Judge Wilson gave his decision in regard to lines 
and land marks, as follows : 

The best evidence is, 

1st. Living marks, such as trees, &c, the first 
and most substantial land marks; and if marked 
trees should not be in a right line, yet the line 
must be run from one marked tree to the next, 
and thence to the next, and so on. 

2d. When there are stones of long standing 
along the line in question, the line must be run 
from the first to the second; from the 2d to the 
3d, &c. 

3d. Old residents in the neighborhood, may de- 
signate marks or points where the original line 
formerly run. 

Lastly, the chain and compass. 



"AZ A MANUAL FOR SURVEYORS. 

It is the practice where there are ditches along 
the line, to take for the line the edge of the ditch 
lying next the bank of dirt thrown out in digging 
it, or if the dirt thrown out in digging be on both 
sides, the middle of the ditch must be taken. 



A MANUAL FOR SURVEYORS. 



23 



PRACTICAL SURVEYING. 



PROPOSITION 1. 



To run a line between two points A and B ; or 
to trace a right line A B on the ground. 

Case 1. When the points A and B can j 
be seen from each other : 

Place the instrument over A, and bring 
the sights to bear on B ; then direct marks 
to be set at the required points m 1 , m*, 
&c, along the line, and it is done. 

N. B. In this and the following propo- 
sitions, I have usedw 1 , n 2 , n s , m\ m 2 , &c, 
to denote the number of the stakes set in 
the lines, reckoning from the beginning. 

Case 2. When obstacles prevent the 
points A and B being seen from each 
other : 

On the same side of A B, and perpendicular to 
it set off equal distances A C and B D ; place the 



B 



24 A MANUAL FOR SURVEYORS. 

instrument over one of these points C, and bring 
the sights to bear on the other point D : then di- 
rect stakes to be set, along the line C D, opposite 
the points required at n\ n*, &c. ; and from these 
set off perpendicular to C D, and equal to A C or 
B D, the distances n 1 m\ n 2 m 2 , &c, towards A B, 
then A m l m 2 B will be the line required. 

Case 3. When the points A and B can be seen 
from an intermediate point E : 

By trials set the instrument at a point E in a 
right line between A and B ; then the intermediate 
points between A E and E B may be found as in 
case 1. 

Case 4. When obstacles prevent A and B be- 
ing seen from an intermediate point, but C and D, 
(case 2d,) may be seen from the intermediate 
point F : 

Set the instrument at F, the intermediate point 
in a right line with C and D, and direct stakes to 
be driven in the line C D as in the preceding case ; 
distances perpendicular to C D, and equal to A C 
or B D being set off towards A B will determine 
the points m\ m 2 , &c., in the line A B. 



A MANUAL FOR SURVEYORS. 



25 



PROPOSITION 2. 

To run a right line on the ground, or to prolong 
the line A B to any distance required : 

Place the instrument over A, the point from 
which the line is to be run. Having adjusted 
the instrument, bring the sights to bear on B, 
a staff placed in the direction of the line to be 
run. Let the needle down on the centre 
pin, and take the bearing of the line, screw 
up the needle carefully ; next remove the in- 
strument to B, set the needle to the course of 
A B and run to C, and from thence to D, and 
so on as far as required. When it is required 
to run the line accurately, proceed thus: Plant 
the instrument at B, and having adjusted it, bring 
the sights to bear on the back station A ; and di- 
rect a stake forward to be set at C ; remove to C, 
and take a back sight to B, then direct a stake to 
be set at a forward station D ; and so proceed as 
far as required. 

This method of running a line by back stakes, 
that is by having a stake set up at the last station, 
and bringing the sights to bear on it, should always 
3* 



13 



26 



A MANUAL FOR SURVEYORS. 



be adopted, unless the lines to be run are very 
short. The needle should not be depended upon, 
except where it cannot be dispensed with, as in 
taking courses, bearings of objects, &c. 

I have used the term "sights," in order that the 
methods pointed out may be applicable to the use 
either of the circumferenter, theodolite or transit. 
Case 2. When there are obstacles which pre- 
vent the line being run directly from A to D : 

Run the line from A to B as before di- 
rected, where we will suppose an obstacle 
p is met with, which prevents the line being 
continued. 

Set off at B, perpendicular to A B any 
s distance, B n sufficient to pass the obstacle 
at B ; from a point P between A and B set 
off another perpendicular P £=B n; pro- 
long t n towards m, until the obstacle at B 
is passed. From r and m in the line t n 
produced, set off the perpendiculars there- 
to, r s=m C=B n from r to s and m to C 
13 prolong s C to D, which will be a point in 
the line A B produced. 



A MANUAL FOR SURVEYORS. 27 

Case 3. When it is impracticable on account of 
intervening obstacles, to lay off perpendiculars t P 
n B, &c, from the line A D : 

Run P t any suitable course ; then run B n pa- 
rallel to it, and make B n = P t. Prolong t n to r 
and m as in the preceding case ; and run r s, m G 
parallel to P t and each of them equal to it ; pro- 
duce s C to D as in the last case, so will D be a 
point in the line A B produced. 

Note. — A line run from a given point which 
does not terminate at the point designed, but falls 
either to the right or left of it, is called a random 
line, or a guide line. 

Case 4. Where swampy ground, &c, prevents 
measuring, as well as running in the direction A B, 
further than to B : 

On arriving at B, deflect the line B C with an 
angle of 60°, that is, make the angle ABC equal 
to 120°, and measure B C. At C deflect C D with 
an angle of 60° ; that is, make the angle BCD 
— 120°, and measure CD. At D deflect the line 
D E with an angle of 60°, or make the angle C D E 
= 120°, and make D E = B C. At E deflect E F 



28 



A MANUAL FOR SURVEYORS. 



k 



with an angle of 60°, or make the 
angle DEF= 120°. The points E 
and F will be in the line A B pro- 
duced. 

Make B G = B C and join C G, 
the triangle B C G is equilateral; 
also C G and D E are parallel, there 
fore D E G C is a rhoniboides, and 



jjijjl E G is equal to C D. Consequently, 
A E is equal to the sum of the dis- 
tances A B, B G, G E, which are 
equal to A B, B C and C D added 

together. 



PROPOSITION 3. 

To measure an angle ABC accurately, where 
BC deflects but a few degrees from the line AB: 

From a point D opposite to A, a point in one 
of the lines, including the angle to be measured, 
run the line DEF till we arrive at the point F, 
opposite C a point in the other line B C including 
the required angle. At E opposite the angular 
point B, measure the perpendicular B E, also mea- 



E 



A MANUAL FOR SURVEYORS. 29 

sure the perpendiculars A D and F C ; 
and also measure D E and E F. 

Draw A g and C h parallel to D F. 

Then B #=B E— E g=B E— A D 
B A=B E— E h=B E— C F 
Nat. Tang, angle B A g=B g -*- Kg = 
B g -7- D E. Also, nat. tang, angle 
B C h is equal toBA-rC^ = BA-f 
EF.* 

The sum of the angles B A g and 
B C h subtracted from 180°, gives V 
ABC. 

This method of finding the angle 
A B C is applicable to measuring little 
deviations from right lines in streets, 
&c, where the utmost precision is re- 
quired that the buildings may be regu- 
lated exactly to the line of the street. 
A transit or theodolite, should be used in this case 
to run out the line D E F, and the distances AD, B 

* If C F = A D we may say E F : F D : : V# AB: V 

h C B the angles being very nearly reciprocally proportional. 



F 



30 A MANUAL FOR SURVEYORS. 

E, C E, &c, should be measured to the hundredth 
part of a foot. 

It has not been thought proper to introduce me- 
thods of measuring angles with the instruments 
which have been described, as that has been pretty 
fully done in the works to which I have already re- 
ferred. My object in the present performance is 
to supply what has been omitted. In doing this, 
I have introduced but little to be found in the au- 
thors alluded to. It may not be improper to intro- 
duce here the method of verifying the correctness 
of an angle by the principle of repetition. Place 
the transit over the angular point B (see the pre- 
ceding figure,) after having adjusted it for obser- 
vation, set the vernier to zero, and bring the tele- 
scope to bear on the staff at A in one of the lines, 
including the angle to be measured ; clamp the 
lower plate to the tripod, unclamp the upper plate, 
and bring the telescope to bear on C a staff in the 
other line comprehending the required angle ; the 
vernier will point out the measure of the angle 
ABC. In this position clamp the plates together, 
unclamp the lower plate from the tripod, and bring 



A MANUAL FOR SURVEYORS. 31 

the telescope to bear on A, by revolving the instru- 
ment bodily on its axis. Now clamp the lower 
plate; unclamp the upper plate, and bring the 
telescope to bear on C ; the vernier will show twice 
the angle ABC. We may repeat this operation 
at pleasure, the last reading of the vernier being 
divided by the number of times the angle had been 
measured, will give a mean result more to be relied 
on than any single observation, however carefully 
made. 

It is best to repeat the measure of an angle, that 
the vernier may pass over the entire circumference 
of the graduated plate, in which case 360° must be 
added to the last reading of the vernier, previous 
to dividing by the number of observations. 

PROPOSITION 4. 

Random Lines. 

To run a right line between two given points, 
when several intermediate stations have to be taken 
on account of intervening obstacles. 

Case 1. When the line can be run from the 
place of beginning : 



32 



A MANUAL FOR SURVEYORS. 



5 

7i I 



*nL 



E 



Having adjusted the instrument at A, 
the point from which the line is to be 
run, and directed the sights towards B, 
as nearly as can be guessed at, this be- 
ing the point to which the line is to be 
run from A, direct stakes to be set at 
every 20 perches (or otherwise, as may 
suit the nature of ground,) in the di- 
rection of the sights, which let be de- 
signated in the order in which they are 
placed by ft 1 , ft 3 , ft 3 , ft 4 , which being 
| m continued until we arrive at C, so that 
f C B may be perpendicular to A C. 
The line A C in the direction of the sights, must 
be carefully run by Proposition 2d, Case 1st, 
Measure C B. Let ft 1 m\ m* ft 3 , ft 3 ra 3 , &c, be 
drawn parallel to C B. The triangles ABC, 
Am 1 ft 1 , Am 2 ft 2 , &c, being similar we have, 
A C : A ft 1 : : B C : »* m 1 
A C : A ft 3 : : C B : ft 3 m 3 , &c. 
AC:AF::CB:FE. 
That is, as the whole distance measured from 
A to C, is to any part of the line measured A n\ 
or A F ; so is C B the distance from the termi- 



A MANUAL FOR SURVEYORS. 33 

urination of the random line, or line run from the 
point at which it should have terminated, to the 
distance from (the line measured,) the points, 
m 1 m 3 , E, &c, in the true line A B. 

Having found n 1 m\ we have n 2 m*, equal to 
twice n 1 m 1 ; also, n 3 m 3 , equal to three times n 1 m 1 , 
&c ; for as the stakes are equidistant from each other, 
the several offsets are some multiple of the first. 

1. As an example, suppose it is required to run a 
line between two farms having a stone at which to 
commence, and also another to which to run. Hav- 
ing run the random line as directed, and leaving 
stakes at every 20 perches distance from the begin- 
ning of the line, A C was found to be 160 perches, 
and B C 2.4 perches ; it is required to find the dis- 
tances to be set off from the random line at every 
stake left in the line, as also the distance to be 
set off at 96 perches from the place of beginning, 
As A C : A^ 1 : : B C : n 1 m 1 
160 : 20 : : 2.4 : .3 
n 2 m 2 = .6; n 3 m 3 — .9 ; n* m* = 1.2 ; n b m 5 — 
1.5 ; n e m 6 = 1.8 ; n 7 m 7 = 2.1 perches. 
4 



34 A MANUAL FOR SURVEYORS. 

AsAC:FA::CB:FE 

160 : 96 : : 2.4 : 1.44 offset at 96 per. 
These distances being severally laid off from the 
random line, to the right or left, according as B is 
to the right or left of C, will give points in the 
line A B as required. 

2. Being requested to run a line between two 
neighbors, I ran a random line 146 perches, and 
found I had missed the true corner by 1.23 
perches. What perpendicular offsets must be laid 
off from this random line, that stakes may be set 
at every 10 perches from the beginning of the line. 

Beckoning from the beginning the several dis- 
tances will be .08 ; .17; .25; .34; .42; .50; .58; 
.67; .76; .84; .93; 1.01; 1.09 and 1.18 perches. 
In general tenths of a foot will be sufficiently exact. 

3. Required the perpendicular distances to be 
set off from a random line 647 perches in length, 
terminating at the perpendicular distance of 47.3 
feet from the point at which the line in question 
must end, the stakes in the random line being set 
at every 40 perches. 

Result, 1st disk 2.9; 2d, 5.8; 3d, S.S; 4th, 11.7; 
5th, 14.6; 6th, 17.5; 7th, 20.5; 8th, 23.4 ; 9th, 



A MANUAL FOR SURVEYORS. 35 

26.3; 10th, 29.2; 11th, 32.2; 12th, 35.1 ; 13th, 
38.0; 14th, 40.9 ; 15th, 43.9, and 16th, 46.8 feet. 

In general it will be most convenient to set the 
stakes in the true line in a retrograde order, begin- 
ning with the last stake in the random line, and 
returning to the first. 

To ensure accuracy in our calculations, we must 
find the offset corresponding to the distance from 
the last stake to the end of the line, which being 
applied to the offset at the last stake in the random 
line will, if all the calculations are correct, make 
up the whole offset C B from the true line. Take 
the last example : — say as 647 : 7 : : 47.3 : 5, 
which, added to the 16th offset 46.8, gives 47.3 ft. 

The following method of finding the offsets may 
sometimes be used instead of the preceding, or 
may be used to verify their correctness. 

Find the proportional part of the offset for the 
distance of the last stake from the end of the line, 
subtract this distance from the whole offset C B, 
and divide the remainder by the number of equi- 
distant stakes in the line, the quotient will be a 
number to be continually subtracted from the last 
offset to obtain the preceding one. 



36 A MANUAL FOR SURVEYORS. 

4. Suppose a random line to be run 242 perches, 
stakes being set in the line 40 perches apart ; what 
distance must be set off at each stake so as to give 
us the true line, the corner being to the right of 
the random line 12.1 feet. 

First.— As 242 : 2 : : 12.1 : .1 correction for 2p. 
From 12.1 take .1 the remainder 12 being divided 
by 6, the number of stakes in the line, gives 2 feet, 
the difference of the offsets at each succeeding 
stake. 

Offset at the end of the line, 12.1 ft. 
" to be deducted for 2 per. .1 



at the 6th stake, 


12.0 


difference of offsets, 


2. 


at the 5th stake, 


10. 




2. 


4th " 


8. 




2. 


3d " 


0. 




2. 



2d 



A MANUAL FOR SURVEYORS. 



37 



Offset at the 2d stake, 



1st 



:£ at the beginning, 



4. 

2. 



2. 
2. 



0. proof. 



The calculation made by the former method is 
as follows : 



As 242 : 40 : : 12.1 : 2. the correction from 


ike to stake. 




Correction at the 1st stake, 


2. 


2d « 


4. 


3d " 


6. 


« 4th " 


8. 


5th " 


10. 


6th " 


12. 


" for 2 per. 


.1 


for 242 


12.1 



It will be sufficiently exact in many cases to 
4* 



38 A MANUAL FOR SURVEYORS. 

omit the fractions of a perch in the first and second 
terms of the proportions, the offsets being small, 
compared with the line of survey. 

5. Run aline 719 perches, when I found I was to 
the left of the true line 3.7 per. What distance 
must be subtracted from the 3.7 offset to obtain the 
offset at 700 perches, and also what number must 
be successively subtracted from that offset, to give 
the distance to be set off at stakes set at every 100 
perches along the line. 

Proportional part for 19 per. is .1 
Offset at the 700 is 3.G 

Number to be deducted for 100 per. 

3.6h-7= .514 per. 

Offset at the 6th hundred, 3.08 or 3.1 



a 


5th 


a 


2.57 " 2.6 


u 


4th 


U 


2.06 " 2.1 


u 


3d 


u 


1.54 " 1.5 


u 


2d 


u 


1.03 " 1.0 


<< 


1st 


a 


.52 " .5 



The following rule is that generally given in 
books on surveying, for determining the true from 



A MANUAL FOR SURVEYORS. 39 

a random line : From the given point or place of 
beginning, run a random line by the given course 
of the line, and measure the perpendicular distance 
between the line so run and the sought corner ; 
then, 

As the length of the line run, 
Is to the said perpendicular distance, 
So is 57.3 degrees, or 3438 minutes, 
To the difference of variation or correction of 
the course, 
Which, being applied to the given bearing, will 
give the present bearing of the line. 

Then set the instrument at the place of begin- 
ning, and run the line by its present bearing. 

This method cannot be relied on, as the short- 
ness of the needle, its aberrations, diurnal varia- 
tion, &c, may lead to error. 

The following method of finding the difference 
of variation is convenient and easily remembered : 
To the length of the measured line add its half 
length, then say, 

As that sum is to the perpendicular distance, so 
is 86° to the correction of the course. 



40 A MANUAL FOR SURVEYORS. 

If we have a table of natural tangents, divide 
the perpendicular distance aforesaid by the length 
of the line, the quotient is the nat. tang, of the 
angle of correction ; take out the angle correspond- 
ing in the table, which will be correct in all cases. 

We may use the nat. sines instead of tangs, as 
far as 5°. 

Example 6th. Suppose a line some years ago 
bore N 40° W 170 per., and that in running by 
this course, we came out 1.55 per. to the left hand 
of the true corner : what is the present bearing of 
the line. 

By the first rule : 

As 170 : 1.55 : : 3438' : 31' correction. 

By the second method : 

As 255 : 1.55 : : 86° : 31'. 

By the third : 

1.55-5-170=. 00912, tang, of 31'. 

Hence 40°— 31'=N 39° 29' W, is the present 
bearing of the line by which it may be retraced by 
the circumfercnter. 

Or place the transit at the beginning of the line, 
and adjust it for observation ; set the vernier to 



A MANUAL FOR SURVEYORS. 41 

zero, and bring the telescope to bear on a stake in 
the random line ; move the vernier 31' to the right 
hand, because the true line is on that side ; the 
telescope will now be in the direction of the line 
to be run out, which may be done as directed in 
Proposition 2d. 

Rule 4th. Multiply the feet offset by 208, and 
divide the product by the length of the line in per- 
ches, the result will be the correction in minutes. 

Rule 5th. Multiply the feet offset by 100, and 
divide the length of the line in 2 pole chains ; to the 
quotient add its -£ T for the correction in minutes. 

If a nonius compass be used : Set the nonius 31' 
to the right hand, the needle being set to N 40° W, 
the sights will give the direction of the required line. 

This is a great advantage the nonius compass has 
over the common circumferenter ; for having set the 
nonius, and clamped it to the difference of variation 
between the present time, and the time a line was 
formerly run ; all the lines of survey run at that 
time may be retraced, by setting the needle to the 
given bearings of the lines. This cuts off the pos- 
sibility of errors arising from applying the differ- 
ence of variation by addition or subtraction to the 



42 A MANUAL FOR SURVEYORS. 

several bearings of the lines ; that allowance being 
made by the nonius, it being the same on all the 
lines of survey, if they have been truly taken. 

Example 7th. — In running a line which, some 
years ago, bore N. 22° 17' E. 311.7 perches, I 
found the true corner 4.5 perches to the right hand, 
what is the present bearing of the line ? 

Am. N. 23° V E. 

Example Sth. — A line being run by a former 
course S. 12° 19' E. 128.7 perches, the corner was 
found 2.3 perches to the right, what is the present 
bearing of the line ? 

Am. S. 11° 18' E. 

Example 9th. — Being called upon to run the 
line between two townships, the course and distance 
of which were given S. 38£° W. 1294 perches, I 
found, in running by this bearing, that the true 
corner was on my left 7.12 perches, what is the 
present bearing of the line between the townships ? 

Am. S. 38° 26' W. 

Example 10th. — Wishing to run a line between 
two points from one of the points I run a course 
IS 89J E, and measured the distance with a two 



A MANUAL FOR SURVEYORS. 43 

pole chain, 129 chains 36 links, when I found the 
perpendicular distance from the line run, to the 
point designated, was 17 feet 5 inches to the left 
hand. Required the course between the required 
points and the several offsets to be set off from the 
line run to the true line, the stakes being 50 per- 
ches asunder. 

Ans. The course is N 89° 31' E. 

1st offset 3 feet 4J inches ; 2d, 6 feet 8| inches. 
3d, " 10 feet 0£ inch; 4th, 13 feet 4f " 
5th, " 16 feet 9 inches ; for the remaining 

part of the line (9.44 perches) 8 inches. 
11th. What allowance must be made on a course 
S 21° E, distance 727 feet, the perpendicular dis- 
tance from this line to the point required being 17 
links. 

Ans. Cor. 53'. The course will be S 20° 7' E. 

Case 2d. When obstacles prevent running the 
line directly from the point A, the beginning of the 

line A B : 

Choose a point C near A, so that A C may be 
perpendicular to A B. 

Bun the random line C E as before directed, 
setting equidistant stakes at n 1 n 2 n 3 , &c. 



44 



A MANUAL FOR SURVEYORS. 



Measure E B : from which 
deduct B D (= A C) the re- 
mainder will be E D. Then, by 
the last case determine the off- 
sets n 1 m\ n 2 m 2 , &c, to each of 
which add A C, and we obtain 
the offsets n l p\ n 2 p 2 , &c, to 
be laid off from the random line 
C E to the true A B. If it be 
required to determine H a point 
at a given distance from A, we 
t may say ; As C E : C F : : E D : 
F Gr, to which apply A C 
(= G H) to obtain F H : whence 
H is determined. As in the 
preceding case we may find the 
angle E C D which applied to 
the bearing of C E, will give the 
bearing of C D, that is of A B, 
because A B is parallel to C D. When E B is less 
than A C, the corrections n l m\ n 2 ??i 2 , &c, must 
be subtracted from A C to obtain the corrections 
for the points p x p 2 , &c, in the line A B. 




A MANUAL FOR SURVEYORS. 45 

Example 1st. Being required to run a line be- 
tween two points, A and B — I measured a perpen- 
dicular A C, equal to 20 feet, and from C run the 
line C E, S 29° W 130 per., when I found the dis- 
tance E B was 27 feet to the right of C E ; what 
is the bearing of A B ; also what distances must be 
laid off from C E to determine points in the line 
A B at 40 per. apart. 

Arts. The bearing is S 29° 11' W. 

1st offset, 22. 1 ft. 2d, 24.3 ft. 3d, 26.5 ft. 

Example 2d. Run the random line C E — S 30° 
W 125 per., and measured A C = 20 ft. and E B 
= 9 ft., the true line being on the left of the ran- 
dom line. What is the bearing of A B, and what 
offsets must be measured off to fix points in A B 
40 per. apart. 

Ans. The bearing is S 30° 18' W. 

1st offset 17.48; 2d, 12.96; and 3d, 9.44 ft. 

Case 3d. When the random line crosses the true 
line, between the extreme points. 

Let A B be the line required to be run, C D the 
random line, C and D being on opposite sides of 
5 



46 A MANUAL FOR SURVEYORS. 

the line A B. Suppose A B 
or C D measures 165 perches, 
A C = 15 feet, and B D 
= 25 feet. As in the pre- 
ceding cases, let stakes be 
driven into the ground in the 
line C D, equidistant from 
each other (say 40 perches.) 
It is evident that the devia- 
tion of the random line C D 
from parallelism with A B is 
equal to the sum of the dis- 
tances A C and B D, or 
15+24 = 40 feet. There- 
i ' fore, in this case we must 
k take the sum of the distances 
A C and BD to find the corrections. 

AsCD:Cw::AC + BD:n* m l = (A C— C F) 

165: 40:: 15 +25 :9.7 = A/ 
Or As C D : w* B : : A C + B D : B D— n* m* 
165: 5 : : 40. : 1.2 =Bg 
15 AC = 15 feet. 




A MANUAL FOR SURVEYORS. 47 

AF = 9.7 



The 1st offset n l m l 



5.3 

9.7 



2d, do. 


n* m* 


= 4.4 

9.7 


3d, do. 


n z m 3 


= 14.1 

9.7 


1th, do. 


n* m 4 


= 23.8 




B<7 


1.2 




BD 


25 



25 proof. 

We cannot subtract 9.7 from 5.3 the first offset, 
which shows the random and true line intersect 
between the points, the difference 4.4 is laid off 
on the other side of the random line, as well as 
the rest of the offsets to the end of the line. The 
point I, the intersection of the lines may be found 
as follows : 

AsAC + BD:AC::CD:CIorAI 
40 : 15 : : 165 : 61.9 

Also AC + BD:BD::CD:DI, orBI 
40 : 25 : : 165 : 103.1 



48 A MANUAL FOR SURVEYORS. 

The correction of the bearing is found as in the 
preceding cases. 

40 feet = 2.42 perches. 
2.42 -f- 165 = .01467 nat. tang, of 50£' this ap- 
plied to the bearing of C D, will give the bearing 
of A B. In this case the offsets decrease at the 
beginning of the line, the correction for the dis- 
tance between the equidistant stakes, must be 
subtracted, until the remainder is less than the 
correction ; then subtract this remainder from the 
said correction, the last remainder will be the dis- 
tance of the offset at the next stake, to be laid off 
on the opposite side of the random line to a point 
in the true one ; the random and true line having 
intersected each other between the stakes, at one 
of which the offset was laid, on a side of the 
random line, different from that of the other ; after 
which the offsets are all laid off on this side to 
the end of the line. Those acquainted with the 
nature of plus and minus quantities, will readily 
perceive the reason of all this. 

Example 2. Given A B or C D = 125 perches. 



A MANUAL FOR SURVEYORS. 49 

A C = 7.7 feet ; B D = 6.2 feet required the off- 
sets, the equidistant stakes being 20 perches apart. 
Arts. The 1st offset is 5.5 feet.") 



2d, 


do. 


3.3 


► To the left. 


3d, 


do. 


1.1 J 




4th, 


do. 


1.1 


To the right of 


5th, 


do. 


3.4 


► the random 


6th, 


do. 


5.6 


line. 



The correction of the bearing is 23'. 
The lines intersect 69.3 perches from A. 

proposition 5. 
Prolonged Lines. 

A 

To determine a point B in the line, A E 
produced. 

Case 1. When the instrument can be placed 
at E, from which A is visible. E 

Direct the sights to A, and prolong the 
line towards B, which continue as far as re- 
quired. 

B 

Case 2.. When obstacles prevent A from being 

seen from E. 

5* 



50 A MANUAL FOR SURVEYORS. 



Measure off a convenient distance F E, 
perpendicular to A B ; make A C = E F, 
then project C F to D, a point perpen- 
dicular to A B from the point B, and make 
B D = E F ; the point B will be in A E 
produced. 



Case 3. When obstacles prevent A being seen 
from cither E or F. 

Run the line A D by proposition 2d. 
At a point F, measure the perpendicu- 
lar F E ; also measure the distances 
A E and A B. Then A E : A B : : 
F E : B D, which being laid off from 
the line A D, gives the point B in A E 
produced. 

The angle DAB may be found by 
the rules given in Proposition 4. 
As A E : F E : : 34 38' : V F A E 
Or | A E : E F : : 86° : V E F : F A E 
Or F E -*- A E - Nat. Tane. V FAE. 



A MANUAL FOR SURVEYORS. 



51 



Place the instrument at A and make the l_ F A E 
as found above ; then run out A E B. 

Case 4. When it is impracticable to run from A, 

CL___4A 

Set the instrument at C, a point 
perpendicular to A B, opposite to A. 
Run a line CGHto H, opposite to 
B. Measure G E, from which deduct 
A C = F E, the remainder is G F. 
Then, as in the preceding case, as 
AE:AB::GF:HD, to which 
apply B D = A C, we get H D + n 
A C = H B. 

Case 5. When G E is less than A C. 




From A C deduct G E, the remainder 
is F G. Then, G C : C H : : F G : 
D H and H B = B D— D H = A C— 

D H. So the point B is determined. 



Case 6. When the random line C H crosses the 
line A E prolonged, 




52 



A MANUAL FOR SURVEYORS. 




In this case, F G = A C— G E 
C G : C II : : F G : D H. 

D H being greater than A C, their 
e difference is B H, which must be laid 
off to the contrary side of C H, to 
which A C was laid ; that is, A and 
B arc on opposite sides of C H, as 
D H : A C : : C II : C I. The point 
of intersection of the random and true 
lines may, therefore, be exactly designated. 

Case 7. When the random line crosses the true, 
between A and E. 

In this case, G E will be on the op- 
posite side of the line A E, to which 
A C is ; therefore, 

F G = FE + E G = A C + E G 
AsCG:CH::FG:DH 
HB = D II — DB = D II — AC 
A and B are on contrary sides of C H. 
I, the point of intersection is found as 
in Case 6. The > D C II is found by 
Case 3. 




A MANUAL FOR SURVEYORS. 53 

PROPOSITION 6. 

To retrace the lines of a Survey. 

Case 1. When the angular points are established, 
run the lines from one angular point to another, 
by the methods already given. 

Case 2. When the angular point is known to be 
in a right line with two other points, and at a given 
distance from each of them. 

Let A and B be the given points, 
C the point in a right line between 



D 



them, which is to be determined in i>- 

order to trace the line D C. B 

Run the line A B as before directed, from A to- 
wards C, measure the distance the point C is 
known to be from A ; also, from B to C, measure 
the distance C is known to be from B, then, if 
these distances both terminate at C, the point is 
determined. The line C D may then be run out 
by the methods already given. But if the mea- 
sures from A and B do not terminate at the same 
point C, which in practice will often be the case, 
measure the given distances from A and B towards 
C ; the distance measured from A, terminating at 



54 A MANUAL FOR SURVEYORS. 

a ; that from B at b. Then it will be, as the given 
distance of A from B, is to the given distance 
AC; so is the distance a b, to the correction a C, 
or as A B : B C : : a t> : b C the correction on B b. 
The correction a C applied to A a or b C applied 
to B b, will determine the point C. 

This method is also applied," when there is not 
full measure. 

Example. Given A C = 75 perches ; B C = 150 
perches, but in measuring these distances, there is 
found an excess of measure, a b = .6 perches, re- 
quired the corrections. As 75 + 150 : 75 : : .6 : 
a C = .2 ; or 225 : 150 : : .6 : b B = .4. 

Therefore, AC = Aa + aC = 75 + .2 = 75.2 
perches, BC = B6 + 5C = 150-f.4 = 150.4. 

Example 2d. Given A C = 325 perches, B C = 
175, and a b = 1.5 perches, excess measure, what 
are the true distances ? 

A C = 325.975 BC = 175.525. 

Example M. Given A C = 120 chains ; B C = 
80 chains, and a deficiency of full measure, a b = 
20 links. What is the lengths of A C and B C by 
this measure ? 



A MANUAL FOR SURVEYORS. 55 

Ans. A C = 119.88 chains, and BC = 79.92 
chains. 

Case 3. When the adjacent lands A C D and 
BCD have been surveyed at very different periods 
of time, measure several lines of the land adjacent 
to A C D, and say as the sum of the lines mea- 
sured, is to the line AC; so is the gain or loss 
of measure on the lines measured, to the gain or 
loss of measure to be applied to A C. In like 
manner find a correction of the measure of B C, by 
measuring several lines of the adjacent land BCD. 
Use these corrected distances for A a and B b in 
the preceding case. 

Case 4. When there are line marks in the line 
D C at D and E, and also in the line A B at B 
andF. 

Prolong the line B F towards 

A, by the method already de- 
scribed ; also prolong D E until 
it intersects the prolonged line 

B F in C, which will be the point sought. 

Case 5. When there are marks only at D and B, 
runs out ome of the lines on the adjacent lands, 



56 A MANUAL FOR SURVEYORS. 

•which are nearly parallel to the lines to be run, by 
which the difference of variation is obtained, which 
being applied to the former bearings, gives the 
present bearings of the lines. The lines may then 
be run out. This may be done with both adjacent 
tracts of land, and a mean of the results taken. 

A reason for selecting lines nearly parallel to 
the line to be run, is that the difference of bearings 
of a line as shown by two compasses, will be the 
same on lines nearly parallel to it. When this 
difference is applied to lines at nearly right angles 
with it, a considerable difference will very often be 
found, which frequently leads to error, unless 
carefully guarded against. 

Case 6. When there is given the bearings and 
distance of the line A B, running from the point 
A, and the year in which it was run. 
A The only method likely in this case to ap- 
proach a satisfactory result, is the following : 

If there have been a line or lines in the 
neighborhood, run the same year, (or there- 
abouts) go to the premises and run them out, 
by which you get their present bearing, and there- 
fore the difference of variation between the present 



A MANUAL FOR SURVEYORS. 57 

time, and the time at which the surveys had been 
made ; this difference allowed on the bearing of 
the line to be run out, will give its present bearing 
by which to run it out. 

If, however, no survey had been made of any 
lands in the neighborhood, by which the difference 
of magnetic variation may be found ; then, in such 
case, if the annual rate of increase or decrease in 
the magnetic declination be satisfactorily known, 
we may ascertain the change of variation in the 
interval of time which applied to the given bearing 
of the line to be run, we shall have its present 
bearing. The variation at West Chester, in 1845, 
is 4° 2' W. 

The variation of the magnetic needle in declina- 
tion, is subject to much irregularity, in some in- 
stances increasing, in other decreasing, and some 
years having scarcely a perceptible motion. The 
annual variation at Philadelphia, has been stated 
at 1° in twenty years. In the neighborhood of 
West Chester, it is about 1° in sixteen years, in 
Warminster Township, Bucks County, 1° 3 r in 
fourteen and one-third years. At any place there 
6 



58 A MANUAL FOR SURVEYORS. 

is much irregularity in a lapse of years. It must, 
therefore, be a matter of uncertainty whether we 
have the correct bearing of the line, even when the 
change for years has been ascertained with the 
utmost care. 

Another source of error in this case, the diurnal 
variation, may be properly mentioned here. If a 
survey be commenced early in the morning, which 
is not completed until one or two o'clock, P. M., of 
a very warm day, it will be found that the bearing 
of the first line of survey will vary several minutes, 
sometimes a quarter of a degree from its bearing 
in the morning. In the winter season, this differ- 
ence will seldom exceed five minutes of a degree, 
but in very warm weather it may amount to fifteen 
minutes. There will be little difference in cloudy 
weather. Surveys should, therefore, as far as 
practicable, be made in the cool part of the day. 
A line which is to be established from the course 
only, should be re-run at nearly the same season 
of the year, a day chosen of much the same tempe- 
rature, and the same time of day, in order to en- 
sure the nearest approach to accuracy the case will 
admit of. 



A MANUAL FOR SURVEYORS. 59 

Other sources of error are the eccentricity of the 
compasses used, the difference of polarity or direc- 
tion of the needles used, &c, all which should be 
carefully guarded against. If the surveyor when 
running old lines were to note the difference be- 
tween the bearing now found and that given, by 
applying this difference to the variation of his nee- 
dle, he may determine very nearly the magnetic 
variation at the time of the former survey. 

A collection of observations of this kind would 
enable him to ascertain the rate of increase or de- 
crease of the variation of the magnetic needle; 
and would be highly valued by those who may be 
investigating this perplexing subject. 

Case 7. When the angular point C is to be de- 
termined from the distances A C and B C only, 
the points A and B being known. 

Measure A C as nearly as may be in 
the direction of C, and at the end of 
the distance, set two stakes a few feet 
apart, so that a line joining them may 
be at right angles with A C. Also, 
measure B C, and set two stakes a few 
feet apart at right angles to B C. Then 



60 A MANUAL FOR SURVEYORS. 

a line joining the former two stakes will intersect 
a line joining the latter two in the point C, the 
angular point required. 

Case 8. When C B bears but a few degrees 
from A C. 

Prolong A C towards D. With the 
angle of deflection B C D as a course, and 
B C a distance, enter the traverse table, 
and take out the latitude C D, and make 
D B perpendicular thereto, and equal to 
the departure ; the point C and the line 
/ ; BC are determined by making D C = 

r° the latitude from the traverse table. 
Case 9. When local attraction affects the needle 
on the several lines. 

Set the instrument at A, direct a stake 

to be set at B, let the needle down on its 

centre pin, and direct the sights to B. 

If the needle does not show the same 

bearing as formerly, move the vernier 

plate till it does. Remove to B, direct 

«^ the sights to A. If the needle does not 

show the same course for B A as at A, there is 

local attraction. Move the vernier till the needle 



A' MANUAL FOR SURVEYORS. 61 

shows the reverse of the given bearing of A B. 
The needle set to the given bearing of B C will 
give its direction. Xext remove to C, and take a 
sight to B, move the vernier until the needle gives 
the reverse bearing of B C. The needle set to the 
bearing of C D will give its direction ; and so pro- 
ceed. Otherwise, apply the angle A B C to the 
bearing of A B. to get the bearing of B C : apply 
the angle B C D to get the bearing of C D. he. 
This may be very accurately done with a theodo- 
lite or transit. 

PROPOSITION 7. 

On Distances, d-c. 

Case 1. Through a given point to run a line 
D C at right angles with a given line A B. 

In the line A B choose two a 

stations, E and F such that the 
angle EDF may be less than 
10°, the point C falling be- *, 
tween E and F. 

"With an instrument mea- 
suring angles to minutes, mea- 

5* 




62 A MANUAL FOR SURVEYORS. 

sure the angles E F D and F E D, the comple- 
ments of which give C D E and C D F ; also, mea- 
sure E F ; then, as the sum of the angles CDE 
and CD Fin degrees and minutes, is to either of 
them, as CDF in the same measure, so is the 
base E F, to the part C F of the base, correspond- 
ing to the < F D C, from which the point C is 
determined. 

If V C D E be used in the above proportion, we 
get E C ; or correctly, as the sum of the nat. tangs, 
of the angles CDE and C D F, is to the nat. tang, 
of either of them, so is E F the sum of the seg- 
ments E C and C F, to the segment corresponding 
to the angle used in the second term of the propor- 
tion. 

Otherwise, subtract the bearing of A B from 
90°, the remainder changing N. to S. or S. to N., 
is the bearing of C D ; then, by trials make C D 
this course ; and C will be the point required. 

Case 2. From the point C in a right line A B, 
to trace a line C D at right angles with it. 

This may be readily done, any of the instru- 
ments used in taking angles, or with the cross 



A MANUAL FOR SURVEYORS. 63 

mentioned in the choice of instruments. It may 
also be done with the chain as follows : 

Make A C = B C = 20, 30, or , 

any other number of links less 
than a chain, place one end of A 
the chain at A, and with the 
other end, trace an arc on the &% 

ground ; remove the end of the chain from A to B ; 
with the other describe a second arc, cutting the 
former in D. A line joining D C will be at right 
angles to A B. In the same manner, another point 
E on the other side of A B may be found. 

Or thus, set the compass at C, and take the bear- 
ing of AB. Subtract this from 90°, changing N 
to S or S to N, gives the bearing of C D. 

Another method : make A C = 4 ; with A as a 
centre and a radius 5 describe an arc ; with the 
centre C and radius 3 describe an arc, cutting the 
former in D the point required. 

Any multiples of 3, 4, 5, as 6, 8, 10, or 30, 40, 
50, &c, will form right Vd triangles. 

Third method : place one end of the chain at C, 
the point at which a right angle is to be made, ex- 



64 



A MANUAL FOR SURVEYORS. 




tend the other end to any con- 
venient point E ; with E as a 
centre, move the other end 
from C (the chain being ra- 
dius,) until it crosses the line A B in A ; prolong 
A E towards D, making D E equal to A E. Join 
C D which will be at right angles with A B. 
Case 3. To measure an inaccessible distance A B. 
Make B C at right angles to A B, 
at C any convenient point. Make 
\ CD at right angles to C B, or paral- 
lel to A B. Set a staff at E in the 
line C B, and in a range with A D. 
Measure B E, E C, and C D ; then 
as E C : B E : : C D : A B. 

This proportion will hold good, if 
B C make any angle whatever with the parallel 
lines A B and C D. 

Note.— If B C = C E, then A B - C D. 
Case 4. To determine the distance A F, which 
cannot be directly measured, on account of the 
obstacle between B and E. 




A MANUAL FOR SURVEYORS. 



65 




Trace the line to B, run B C. 
From C run C D parallel to A B. 
From D run D E parallel to B C, 
and make it equal to it. Then run 
E F the same course as A B : F will 
be a point in the line A B continued. 
The distance A F will be equal to 
the sum of the distances A B, C D 
and E F. 

Case 5. To find the bearing and distance of A 
from B, accessible only at its extremities. 

Choose a point from which . c 
A and B are both accessible. 
Prolong A to D, making D 
equal to A ; also, prolong 
B Oto C, making C = B. 
Join C D, which will be equal 
and parallel to A B. Its bear- 
ing and distance is therefore 
determined. 

Case 5. When A and B are inaccessible. 

Plant a staff at C and find the distance A C and 
C B by the preceding methods : 




66 



A MANUAL FOR SURVEYORS. 




Make C D as many parts 
of A C as you do C E of 
C B ; join D E and measure 
it. Then, as C D : C A : : 
D E : A B, or as C E : 
CB::DE:ABDEis 
parallel to A B. 



Case 6. To prolong the line A B across a river, 
&c, and determine the distance across to E. 

At a point C make the an- 
gle B C D a right one, at D 
make the angle B D E a right 
angle. Measure B C and D B. 
Then B C : B D : : B D : 
BE. If D C be measured, 
BC:DC::DC:CE. 

Otherwise. Make^LCDE 
= 45°, then C D will be 
equal to C E, the breadth of the river. 




A MAXEAL FOE SURVEYORS. 



67 



If < C D E = 26° 34' then C E = J C D 
"<CDE = 33°41 / " CE = |CD 
"<CDE = 45° " CE= CD 

"<CDE = 56° 19' " CE=fCD 
"<CDE = 63°26 / " CE = 2 CD 
Or, if C B D = B D E = 60°, the triangle BDE 
becomes equilateral, and B D = B E. 

Case 7tJi. To find the horizontal distance A B 
through a precipice or clift too steep to be mea- 
sured with the chain. 

Plant a staff at C on the top of 
the cliff, aligning it with A and B, 
both of which may be seen from C. 
Run the line A E and measure it. 
Also measure the angles C A E = 
D A E ; and C E A = D E A (the 
point D being directly below C.)t>|jj 
Find, by trigonometry, the dis- 
tance A D. In like manner from 
B run B F and measure it, also 
the angles C B F = D B F ; and 
B F C = B F D to find B D. 
Then A B is the sum of the hori- 




68 



A MANUAL FOR SURVEYORS. 



zontal distances A D and B D. In this solution the 
planes A D C, E D C, B D C and F D C are con- 
ceived to be at right angles to the plane A E D F B. 
By taking the altitude to the top of the staff C from 
two stations on each side of the steep, the horizon- 
tal distance may be found. 

Case 8. To measure an inaccessible distance 
AC. 

Plant a staff at D in a line with 
A C. Run B C and measure it; 
and make E D parallel thereto, and 
measure E D and also C D. Then 
EF(=ED — BC) :B C : : CD 
to A C. This case is applicable 
when B C and E D differ consider- 
ably, that is, E F bears a due pro- 
portion to B F or C D. 
Note. — This Case should have followed Case 3. 




PROPOSITION 8. 

Parallel Lines. 
Case 1. From a given point C, to run a line C D 
parallel to A B. 



A MANUAL FOR SURVEYORS. 



69 



Run the line A B to find its bearing, 
remove the instrument to C and set the 
needle to this bearing, the sights will 
then be parallel to A B, so C D may be 
run out. 

Or set the instrument at B and mea- 
sure the angle ABC. Remove to C, 
making the angle BCD equal to the supplement 
of A B C. Then will C D be parallel to A B. 

If a transit be used after having removed to C, 
reverse the telescope on its axis, bring it to bear 
on B (the vernier being at V A B C,) clamp the 
lower plate, bring the vernier to zero, the tele- 
scope will be parallel to A B ; reverse the telescope 
on its axis, and set a stake at D. 

Case 2. When B is not accessible from C, 

Plant stakes at A, B, C, also 
at E, in a line with B C. Find, 
by the preceding methods, the 
distances A E, E B and E C. 

AsEB:EC::AE:ED. 
This being laid off in A E to D, a 
line joining C D will be parallel 
to AB. 

7 




70 



A MANUAL FOR SURVEYORS. 



Second Method. — Run A C and make <s A C D 
B A C. 



Otherwise. Bisect A C in I. In 
B I produced, make I D = B I. Join 
C D, which will be parallel to A B. 




PROPOSITION 9. 



To determine a point C in a right line with A B, 
the point B being a steeple, or other conspicuous 
object, which is visible from A and C, and inac- 
cessible from both. 

Measure the angles of deflection B A D d D E, 
e E F, D E and F being suitable points for 
measuring the angles of deflection. The station 
F being selected as near C as can be judged upon 
the ground. 

The angles of deflection at D and E being to 
the left, their sum must be diminished by the 



A MANUAL FOR SURVEYORS. 



71 



angle at A : the remain- 
der is the deviation from 
parallelism of the lines 
A B and E F. This re- 
mainder subtracted from 
90°, or a right angle gives 
the angle of deflection 
/ F G at F, to make F G 
at right angles with A B 
prolonged. Bun F C G, 
measuring the distance 
F G. Observe the an- 
gles B F G and F G B, the complements of which 
are F B C and G B C. The sum of these comple- 
ments is F B G. 

As the angle F B G, in degrees or minutes, 

Is to the angle F B C in the same measure, 

So is F G 

ToFC, 

Or, as <, F B G : <; G B C : G F : C G. 

From either of these proportions the point C is 
determined, which will be in A B produced, F G 
is the complementary course of A B. It may be 




72 A MANUAL FOR SURVEYORS. 

run out where there is no local disturbance of the 
needle by setting the instrument at F, and run F G 
by this complementary course. The base F G of 
the triangle B F G should be of such a length, that 
the angle F B G may be only a few degrees ; if it 
should be too great, the distances F C and C G will 
not increase in the same ratio with the angles F B C 
and C B G. 

N. B.— If F B G is greater than 10°, the seg- 
ments F C and G C should be found by the usual 
rules of Trigonometry. 

The following proportion is in substance the 
same as the preceding. From the external angle 
B G H, take the <. B F G; then say, 
As this remainder is to the difference between 
the angle B F G and a right one ; so is F G to 
F C as before. 

It may sometimes be convenient to take several 
stations in deflecting from A to F ; but in all cases 
the angles of deflection to the right hand must be 
added together, and also those on the left of the 
lines deflected ; the difference of these sums will be 



A MANUAL FOR SURVEYORS. 73 

the deviation, from parallelism, of the last line der 
fleeted from the line A B. 

If the deflected angles on the right exceed those 
to the left, the difference must be laid off to the 
left, and vice versa: the telescope will then be 
parallel to A B. 

If at every station we arrive at, we set the ver- 
nier to the same degrees as at the last station, re- 
versing the telescope on its axis or in its wyes, and 
bringing it to bear on the last station point ; then, 
having clamped the lower plate to the tripod, bring 
the telescope in its direct position on the next sta- 
tion, the vernier will perform the additions and 
subtractions of the angles of deflection ; conse- 
quently, when we arrive at any station when the 
instrument is adjusted by the last station point, 
bring the vernier to zero, the telescope will be 
parallel to A B — but if set to 90°, it will be at 
right angles with it. We may make any angle 
whatever with A B, by setting the vernier by 
means of the tangent screw to that angle. 

Given < F B C = 2° 5' ; G B C = 2° 55' and 
7* 



74 



A MANUAL FOR SURVEYORS, 



F G = 12 ft., to find F C (the points F and G 
being determined as above). 
As F B G (3000 : F B C (125') : : F G (12 ft.) : 
F C = 5 ft. 
F B G (3000 : C B G (1750 : : F G (12 ft.) : 
G C = 7 ft. 
Case 2d. When C falls without the triangle 

FBG. 

Find, as before, the angle FBG, 
the angle G B C, which is equal to 
the difference between B G C and 
a right angle, and the distance 
FG. 

As F B G : <. GBC::FG: 
G C, which, being laid off on F G 
produced, determines C, a point in 
A B prolonged. The contents of 
this proposition and Case 1st of 
Prop. 7th, are believed to be new ; 
nothing of a similar character is to be found in any 
publication with which I am acquainted. 

Given F B C = 50' ; G B C=10' and F G = 8 ft. 




A MANUAL FOR SURVEYORS. 75 

to find GC; F G being determined as above di- 
rected. 

As <, F B G (= 50' — W=4&) : G B C (10') 
: : F a = (8 ft.) : G C = 2 ft. 

Given F B C = 83' ; G B C = 17' and F G = 
33 ft. to find G C. Ans. G C = 8.5 ft. 

if G is between F and C ; but if not ; G C = 5.61 ft. 

proposition 10. 

Dividing Land. 

An easy rule for finding the angles of a right 
angled triangle, the sides being given. To the 
hypothenuse add half the longer leg. Then, as 
that sum is to the shorter leg, so is 86° to the 
angle opposite the shorter leg. This rule, which 
is easily remembered, is very useful in many cal- 
culations in the field, where tables cannot be con- 
veniently used. The greatest error does not ex- 
ceed — 4 minutes. The rule is therefore sufficiently 
exact for most purposes in surveying to which it 
may be applied. 



76 



A MANUAL FOR SURVEYORS. 



Example. Given the 3 sides of a right angled 
triangle, 30, 40 and 50, to find the angles : 

(50 + 20) : 30 : : 86°— 36° 53' the less angle. 

Example 2d % Given the hyp. and greater leg of 
a right angled triangle 50 and 40 angle opposite 
less leg 86f°, to find the less leg. 

As 86° : 36 T ° : 70 : 30, as required. 

This rule may be readily applied in cutting off a 
trapezoid from a given tract of land that shall con- 
tain a given number of acres, the angles being 
nearly right angles. 

Let there be given A B 

south, B C west 40 per. and 

C D N. 4° W., to cut off a 

B trapezoid A B C D containing 

5 acres, by a line A D parallel to B C. 

First, 800 -r- 40 = 20 = C E approximate. 

In this case the leg and hypothenuse are nearly 
equal, we may use one for the other, therefore, 

As 86° : 4° : : (20 + 10) :1.4 = ED approxi- 
mate. 

AD = AE + ED = BC + ED=40 + 1.4 
= 41.4 approximate. 



E 

C\ 



A MANUAL FOR SURVEYORS. 77 

Twice the area of a trapezoid, divided by the 
sum of the parallel sides, gives the perpendicular 
distance between them. 

1600 -7- (40 + 41.4) = 1600 -r- 81.4 = 19.66 = 
C E = A B. 

As 86° : 4° : : (19.66 + 9.83) : 1.37 = D E. 

A D = 40 + 1.37 = 41.37 sufficiently correct. 

1600 -v- (40 + 41.37) = 19.66 as before (the 
proof.) 

The angle B being a right angle, no correction 
is required on that side of the trapezoid. 

Another method is to find the area of the trian- 
gle D C E, and cut off a small trapezoid equal to 
it, either within or without, as the case may re- 
quire. In the preceding example E C + |- D E = 
area of C D E = 20 -f- .7 = 14., this, divided by 
A D 41.4, gives .34, and this subtracted from E C, 
20, because A D is greater than C B, gives the 
correct value of E C = 20 — .34 = 19.66, as be- 
fore. 

In most cases the use of the traverse table is 
more expeditious to find D E. Taking the ap- 
proximate value of C E in a lat. column, under 4°, 



78 



A MANUAL FOR SURVEYORS. 



gives, in a departure column, 1.4 = D E, from 
which a correct value of C E is found. With 
19.66 in a lat. column, under 4° in a distance 
column, is found 19.71 = CD. 

Example 2d. Given A B south, B C west 40 
per., and C D N. 4° E. to find A B, C D and A D, 
when the trapezoid ABCD contains 5 acres. 

Arts. A B = 20.36 perches. 
C D = 20.41 " 
A D = 38.58 " 
When it is required to cut off a tra- 
pezoid from a trapezoidal piece of 
land, it may sometimes be done in the 
following manner : 

Given A B N. 40° E. 44.4 perches, 
B C S. 50^ E. 60.8 perches, CDS. 
40° W. 46 perches, and DAN. 49° W. 60.9 per- 
ches, to cut off 5 acres by a line E F parallel to 
AB. 

First.— 800 -± 44.4 = 18 = E A approximate. 
D^ = DC- A B = 46 — 44.4 = 1.6. 
A D : A E : : D g : E h or 60.9 : 18 : : 1.6 : .47. 
EF = FH/iE = AB + E/i = 44.4 + .47 
= 44.87. 



H 



LA 



A MANUAL FOR SURVEYORS. 



79 



A h = F B = 1600 -^- (44.4 + 44.87) = 17.93 
the perpendicular corrected. 

In a lat. column with 17.93 under 1J° = <^AD 
in a distance column, is found 17.94 = A E. 

When calculations are made by J. Gummere's 
rule, (the sides A D and B C, as in the above ex- 
ample, being nearly parallel,) great care must be 
used in extending the decimals to 3 places to en- 
sure accuracy, as two decimals only may throw the 
line E F a pole from its true position in many 
cases that occur in practice. 



proposition 11. 



To determine the correct bear- 
ings of the lines of survey where 
local attraction deflects the needle 
from its usual direction or mag- 
netic position. 

Let A B C D be several sides of H 
a survey on which the needle is 
disturbed by some extraneous mat- 
ter. 



80 



A MANUAL FOR SURVEYORS. 



With the compass or circumferenter : Place the 
the compass at A, take a back sight to X, the last 
station, and note the bearing, then sight to B and 
note its bearing. Having the bearing of A X and 
A B, both from the same station A, we can find 
the angle X A B as correctly as if the needle set- 
tled in its true position, for the needle must be 
equally affected when the bearings were noted. 
Remove to B, and take a back sight to A, noting 
its bearing, then direct the sights to C, and note 
its bearing, from which the angle ABC will be 
correctly obtained. Thus proceed until all the 
angles are taken. If the entire survey has been 
made as above directed, the sum of all the internal 
angles will be equal to twice as many right angles 
as the figure has sides, diminished by four right 
angles. If this sum, as in practice will be likely 
to be the case, should differ a few minutes from 
what it should be, the minutes of error may be dis- 
tributed among the angles by addition or subtrac- 
tion, according as there is defect or excess in the 
sum of the observed angles. 

Now, having all the correct angles, assume some 



A MANUAL FOE SUEVEYOES. 81 

side of survey as X A to be correct, being least 
affected by local attraction ; then applying the an- 
gles severally as they come in order of survey, we 
will have the bearings of the sides as correct, rela- 
tively, as if no local attraction existed. 

If a nonius compass be used, place it at A, take 
the bearing of A B, remove to B, take a back 
sight to A and clamp the sights upon it. Un- 
clamp the nonius plate, and with the pinion and 
rack move the nonius plate until the needle gives 
the reverse bearing of A B, which it had at A. 
Unclamp the sights and bring them to bear on C, 
the needle will show the correct relative bearing of 
B C, which note. Remove to C, take a back sight 
to B, and clamp the sights upon it; move the 
nonius plate by the rack until the needle shows 
the reverse bearing of B C ; unclamp the sights 
and take the bearing to the next station, and so 
proceed till the survey is completed. The relative 
bearings thus obtained will be as correct as if no 
local attraction influenced the needle. 

If a theodolite or transit be used, the internal 
angles may all be measured by the limb of the in- 
8 



82 A MANUAL FOR SURVEYORS. 

strument, without regard to the needle. From 
which, having also the bearing of one line, the 
bearings of all the lines may be found. 

The external angles, or angles of deflection, may 
also be taken as follows : 

Place the instrument over A, reverse the teles- 
cope on its axis or in its wyes, set the vernier to 
zero, and bring the sight to bear on X ; then clamp 
the lower or graduated plate in this position, re- 
verse the telescope to bring it again in its direct 
position, bring telescope to bear on B, (by means 
of the tangent screw or rack,) the index or vernier 
will, being read, give the angle of deflection of the 
lines X A and A B. Remove to B and take the 
angle of deflection to C in the same manner as 
from A to B ; proceed thus the entire circuit of the 
survey. If all the angles of deflection have been 
outward, their sum must, if correctly taken, be 
equal to four right angles, or 360°. If any of the 
angles are re-entering, the sum of the external 
diminished by the sum of the re-entering angles, 
will be equal to 360°. 

If the telescope be reversed, the vernier being at 



A MANUAL FOR SURVEYORS. 83 

the same division as at the last station, the hair of 
the telescope cutting the last station point, then 
clamping the lower plate to the tripod, reverse the 
telescope to bring it in a direct position, bring it 
to bear with the tangent screw on next forward sta- 
tion, the vernier will show the angle of deflection. 
Proceeding thus from station to station, the vernier 
will give the sum of the angles of deflection, and 
hence, when we arrive at the first station, the ver- 
nier will be at zero, or the point at which it was 
placed when the survey was begun. Its distance 
from this point is the sum of the errors in observ- 
ing the angles which may be distributed amongst 
them, so as to make the proper sum or quantity 
for the angles. 

If there should be a considerable difference or 
error it would be advisable to retrace some of the 
lines until the error be discovered. 

CALCULATIONS. 

It is the practice with some surveyors to read 
the bearings of lines to quarter degrees, and note 



84 A MANUAL FOR SURVEYORS. 

the distances in chains and links, which, in calcu- 
lations, thej reduce to perches and hundredths. 

In measuring distances, where the line to be 
measured is one hundred perches or upwards in 
length, it must be evident to those who may be 
acquainted with the ordinary mode of measuring, 
that that distance cannot be measured to the hun- 
dredth of a perch, and frequently not even to the 
tenth. 

Again, when courses are read to the nearest 
quarter degree, there is a probability of an error 
which may reach to half that quantity or 7£', 
which, in the distance of 100 perches, gives 2 
tenths of a perch departure from the point of ter- 
mination. Therefore, when bearings are read to 
quarters of a degree, and distances measured to 
tenths of a perch, it will not conduce to accuracy 
to extend the calculations for the area to hun- 
dredths or another decimal figure. 

The bearings of lines should always be read to 
the nearest five minutes, and distances over 100 
perches need not be more exactly noted than to 
tenths of a perch. 



A MANUAL FOR SURVEYORS. 85 

In laying out town lots, or where the utmost 
precision is desired, angles or bearings should be 
measured to the nearest minute, and distances to 
hundredths of a perch, or tenths, or hundredths of 
a foot. 

For this purpose a theodolite or transit should 
be used to measure the angles, and a twenty feet 
frame, with a level or plummet attached, having a 
slider affixed at either end of the frame, to adjust 
it to horizontal admeasurement. To measure a 
line with the frame, in the first place, the line 
should be "boned," as it is technically termed, 
that is, pegs or short stakes, at the distance of 20 
feet, should be driven in the line nearly even with 
the surface of the ground ; then placing one end 
of the frame at the beginning of the line, the other 
on the first stake, after having adjusted the frame 
to a level, make a fine scribe or mark on the top of 
the stake, precisely at the end of the frame; next, 
bring the frame forward, adjust the hind end to 
the scribe on the stake, bring it to a level and 
scribe the second stake at the end of the frame ; 
and so proceed to the end of the line. 
8* 



86 A MANUAL FOR SURVEYORS. 

A line, several hundred feet in length, if mea- 
sured as above directed, will be within an inch or 
two of the truth. 



METHOD OF OBTAINING THE FIELD NOTES OF A 
TRACT OF LAND ACCURATELY. 

Having given directions in the preceding part 
of this work, best suited to determine correctly the 
position of any line of a survey which may be de- 
sired to be run out or retraced, it may be proper 
here to give an example embracing a number of 
those cases, which occur in practice, so as to ex- 
hibit the application of the rules which have been 
given. 

Let us suppose it be required to survey a tract 
of land ABCDEA with a transit and common 
two pole chain. 

Place the instrument, by means of a plummet, 
exactly over the point A of beginning. After 
having adjusted it for observation, set the vernier 
to zero, and clamp it there ; next, let the needle 
down upon the centre-pin, revolve the instrument 



A MANUAL FOR SURVEYORS. 



87 




so as to bring the needle to the north and south 
points of the compass box, clamp the lower plate to 
the staff head, unclamp the vernier plate and bring 
the telescope to bear on a staff set at B, the second 
station ; the course may be read from either the ver- 
nier or needle, or both, which is preferable. Sup- 
pose the vernier reads 50° 2V and the needle N. 



88 A MANUAL FOR SURVEYORS. 

50° 20' E., also A B measures 68 chains and 47J 
links. The course and distance may be set down 
A B N. 50° 21' E. 137.9 perches, using the course 
shown by the vernier in preference to that shown 
by the needle. The observations at A having been 
finished, remove the instrument to B, adjust it for 
observation ; reverse the telescope, the vernier 
being at 50° 21', the bearing at the last station 
A B, bring the telescope to bear on a pole at A, 
clamp the instrument to the tripod, the zero points 
or north and south points of the graduated plate 
and the compass box will be in the magnetic me- 
ridian, and consequently parallel to its position at 
A ; the vernier will therefore show the bearing of 
a line, the same as the needle. Unclamp the ver- 
nier plate and bring the telescope to bear on a 
third station C. Let the needle settle — suppose it 
reads N. 79° 55' E ; at the same time the vernier 
reads 79° 54'. In measuring this line B C, an 
obstacle, a clump of bushes and swamp prevents its 
being directly measured further than b, 20 chains 
from B. An offset, b a, being measured 12 feet at 
right angles with B C, also at c, a point between 



A MANUAL FOR SURVEYORS. 89 

B and b, a perpendicular offset of 12 feet was made 
from c to d. Prolong the line d a to e, where set 
off an offset ef = 12 feet ; continue the line a e to 
g, make g h = 12 feet, perpendicular to a e, con- 
tinue/ h to C. It is found a e measured 14 chains 
25 links, and / c == 10 chains 21 links, so the 
whole line "B C measures 44 chains 46 links ; the 
bearing and distance of B C is N. 79° 54' E. 89.8 
•perches. 

Place the instrument at C, take a back sight to 
B, the vernier being at 79° 54', clamp the lower 
plate to the tripod, release the upper plate. The 
station D being out of view, run a random line in 
that direction, as nearly as may be, setting the 
vernier to 130° ; the needle reading S. 50° E. at 
the same time, run the line 47 chains to the bank 
of a deep creek at m, on the opposite side of which 
set a stake at p, a point in the continuation of the 
random line C q. Measure a perpendicular m n 
= 6 chains, and make the angle m n p = 26° 34 r ; 
consequently m p is equal the half of m n = 3 
chains. From p continue the random line to q, 
19 chains further ; when we arrive opposite to D 



90 A MANUAL FOR SURVEYORS. 

on the left of the random line, I) C is 69 chains = 
138 perches. The corner D from q is 9.25 feet - 
56 perches; f C D = 207 : .56 : : 86° : 14' the 
correction which apply to C q, gives the bearing of 
C D 129° 46' or S. 50° 14' E. 138 perches. 
Stakes being set in the random line C q at every 
20 perches, the calculation for the offsets is as fol- 
lows : 138 : 20 : : 9.25 ft. : 1.35 ft. The distances 
to be laid off at the several stakes will be 1.35 ; 
2.7; 4.05; 5.4; 6.75 and 8.1 feet, these distances 
having been laid off, establishes the line C D. 

At a point in the random line C q, near q, and 
in the line D E let the instrument be set for ob- 
serving the course of D E. The telescope being 
reversed and brought to bear on a back stake p, 
in the random line C q, the vernier at the same 
time reading 130°, let the lower plate be clamped 
to the tripod, bring the telescope to its forward 
direction, and by means of the tangent screw, 
make the hair cut E or a stake in a range with 
D E, the vernier gives for the bearing of D E 
270° ; the needle due west measuring 7 chains 25 
links from D towards E, a high and steep rock was 



A MANUAL FOR SURVEYORS. 91 

encountered, on the top of which a stake was set 
at r in a right line with D E. At the termination 
s of the 7 chains 25 links, a right angle s t was set 
off from the line D E, equal to 5 chains, making 
the angle s t r = 63° 26', the distance r s must be 
10 chains. From r to E the ground was a regular 
slope. I set my instrument at r, and took the 
depression to E 11° measuring the oblique line r E 
IT chains 25 links. With 11° as a course and 17 J 
in a distance, the latitude is 16.68 or 16 chains 34 
links, equal to 33.36 perches for the horizontal 
distance ; therefore D E measures, horizontally, 34 
chains 9 links or 68.36 perches. The instrument 
being removed to E and adjusted as at the other 
stations, the telescope being directed to A, the ver- 
nier will read 266° 12' ; the needle S. 86° 10' W., 
and measuring the distance A E, it will be found 
to be 116 chains 20 links. The instrument next 

m 

placed at A, reverse the telescope, and bring it to 
bear on E, the vernier being at 266° 12', and clamp 
the lower plate to the tripod. The telescope being 
in its direct position, and brought to bear on B, 
the vernier, if the work is correctly done, will read 



92 A MANUAL FOR SURVEYORS. 

50° 21 r ; this being the point at which the vernier 
was set in first setting out, is the proof that the 
angles have been correctly measured. The courses 
and distances in this example will be as follows : 
A B N. 50° 21' E., distance 137.9 perches. 
BCN. 79° 54' E., " 89.8 
CDS. 50° 14' E., « 138.0 
D E West, " 68.36 " 

E A S. 86° 12'W. 5 " 232.8 
The outward angles by the vernier will be at 
A 50° 21', or 144° 09' < of deflection. 
B 79° 54', or 29° 33' " 

C 129° 26', or 49° 32' 
D 270. or 140° 34' " 

E 266° 12', or 3° 48' " 



Sum of right hand <'s 363° 48' positive <'s. 
" left " " 3° 48' negative " 



Proof, 360° 00' 

Note. — It is sometimes conducive to accuracy to 
measure diagonal lines, or lines to opposite corners 
of the tract surveyed. 




A MANUAL FOR SURVEYORS. 93 

N. B. Angles of a survey may be measured 
with the chain as follows : 

Let A be the angular point, A 
A B the direction of one of 
the lines, and A C the line of 
direction of the other line. 
Measure A B and make A C equal to it, and join 
B C and measure it. The < A in the isosceles 
triangle A B C is readily found. 

Or, A B, B C and A C may all be unequal in 
which case the angles must be found by the rules 
for solving oblique angled triangles. 

It sometimes happens that an old road is requir- 
ed to be straightened between two given points, in 
which case it may be desirable to know approxi- 
mately the course between the given points. This, 
if the courses do not differ much amongst them- 
selves on the old road, may be ascertained as in the 
equation of payments, using the courses and dis- 
tances as the payments and time are used in the 
ordinary rules of arithmetic. 
9 



94 A MANUAL FOR SURVEYORS. 

Given N. 41° E. 20 per. 41° x 20= 820 

N. 43° E. 30 " 43° x 30 = 1290 

K 42° E. 80 " 42° x 80 = 3360 

N. 44° E. 120 " 44° x 120 = 5280 

N. 40° E. 200 " 40° x 200 = 8000 

What is the course of 

the straight line joining 450) 18750 
the extreme points ? 



41|° 
The course is N. 
41|° E., nearly. 
Or thus, taking the several courses from 41°. 
0° x 20 = 
2° x 30 = 60 
l°x 80 = 80 
3° x 120 = 360 
1° x 200 = 200 



450) 300 



41 

N. 41§ E. as before. 
By the traverse N. 41° 35' E. 



A MANUAL FOR SURVEYORS. 95 

Given N. 2° E. 60 perches. 
1ST. 3° W. 90 " 
K 1° W. 80 " 
K 1° E. 120. U 
to find the course of the strait line joining the ex- 
treme points. 

E. W. 

N. 2° E. x 60 = 120 

N. 3° W. x 90 = — 270 

N. 1° W. x 80 = 80 

N. 1° E. x 120 = 120 

350 240 350 
240 

350)110(N. 0° 19' W. 

By the traverse we get N. 0° 19' W. 

There is too much uncertainty in the use of the 
above method, except in particular cases, to make 
it generally useful. The traverse table should 
always be used if at hand. 

It is sometimes required that stakes should be 
set off from the respective angular points, to the 



96 A MANUAL FOR SURVEYORS. 

line joining the extreme points, either to save 
trouble of running a random line or for proving 
the truth of the operations. 

In such case take the difference between each 
given course and the course of the closing line, 
noting whether the given line is to the left or right 
of the closing line, with this difference as a course, 
and the given distance of the line, take the de- 
partures from the traverse table, then the first de- 
parture will be the distance of the closing line from 
the angular point at end of the first distance. The 
sum or difference of this and the next departure, 
according as they are both to the same hand or to 
different hands, will be the distance of the closing 
line from the end of the second distance, and so 
proceed to the last point where the closing line 
and last distance will come together, if rightly 
done. 

Example. Given the bearings and distances of 
several lines as follows, viz : N. 40° E. 50 perches, 
N. 38° E. 24 perches, N. 45° E. 40 perches, N. 
39° E. 100 perches, required the distance of the 
closing line from each angular point. 



A MANUAL FOR SURVEYORS. 97 

The bearing and distance of the closing line will 
be found to be N. 40° 15' E. 213.88 perches. 
Hence (marking right hand + and left hand — ) 

N. E. N. E. p. Lat. Dep. p. 

40° 0' 40o 15' — 15' 50 50. — .22 — .22 B b 

38o « _ 20.15' 24 23.98 — .94 — 1.16 C c 

45o « _j_ 4».45' 40 39.86 3.31 2.15 D d 

390 " — 1°.15' 100 99.98 — 2.18 — 0.03 E e 



213.82 



E e should, when correctly done, be =0 

N. B. The closing line is not exactly N. 40° 15' 
E. ; but offsets computed from this line (N. 40J° 
E.) determine B, C, D, &c, as correctly as if the 
closing line had been used, and vice versa. 

Given A B, N. 10° E. 15 perches, B C, N. 15° 
E. 19 perches, C D, N. 17° E. 40 perches and 
D E, N. 12° E. 35 perches, to find the several 
offsets from the closing line to the angular points 
A, B, &c. 

Ans. The bearing of the closing; line running 
from A, is N. 14° 5' E. 108.85 perches, the offsets 

are 

9* 



98 



A MANUAL FOR SURVEYORS. 



B b = 1.07 perches. 

C c = .77 " 

Dd = 1.26 " 

Ee= .00 " 
Note. — The reverse of the 
above operation ; that is, run- 
ning a straight line from one 
extreme point to another 
along an irregular boundary, 
and by calculation, finding 
how far each angular point is 
from the line so run, may be 
employed to determine the 
angular points in the irre- 
gular boundary, especially 
where obstructions render the 
running on the true line dif- 
ficult. If the irregular boun- 
dary deviates but little from 
a right line, this method is 
the most accurate that can be 
employed. 



A MANUAL FOR SURVEYORS. 99 

To straighten a crooked boundary between two 
estates, so that each estate may have the same 
quantity of land. 

Let A B C D be a crooked boundary. It is re- 
quired to run a straight line from A to a point P, 




in the line passing through D, that shall equalize 
the quantity of land as before. Run any line A E, 
near the boundary, and measure perpendicular 
offsets from this line, to the several bends in the 
crooked line, and find the areas of the several tra- 
pezoids, the sums of which areas will be the area 
of the irregular figure A B C D E A, which being 
divided by the half of A E, will give E P, the point 
P being that to which a line drawn from A will 
equalize the areas as required. 

If the point A should be at a short distance from 
the boundary, its distance must be taken from the 
above quotient or added thereto, as the case may 
require, to obtain E P as before. 



100 A MANUAL FOR SURVEYORS. 

If equidistant ordinates or offsets be taken ; add 
together half the sum of the extreme offsets, and 
the sum of all the intermediate breadths or offsets, 
which, being multiplied by the equal distance be- 
tween the ordinates, the product will be the area 
of the irregular figure as before. Whence the dis- 
tance E F is found as in the former case. 

Example. Let the ordinates or offsets, at six 
equidistant places be 4, 6, 2, 3, 5 and 8, the equi- 
distance apart being 50. 

Here ~f-£ = \f = 6 the J sum of extremes. 

Then (6 + 2 + 3 + 5) = 22 and 22 x 50 
= 1100 area. 

Whence 1100 + (5 ' X 50 -*- 2) = 1100 - 125 
= 8.8, and 8.8 — 4 = 4.8 = E P, therefore, a 
line joining the beginning of the boundary near A, 
with the point P, will fulfill the conditions pro- 
posed. 

VARIATION OF THE COMPASS. 

The irregularity of the variation or declination 
of the magnetic needle in causing uncertainty in 
retracing old lines of survey, is well known. A 



A MANUAL FOR SURVEYORS. 101 

step towards obviating the errors attending the old 
method of finding the difference of variation or de- 
clination, and thereby obtaining the present mag- 
netic bearing of the lines of survey, has been taken 
by the Legislature of Pennsylvania, by enacting, 
that meridian lines should be established in the dif- 
ferent counties of the commonwealth, and that sur- 
veys hereafter made should be returned according 
to the true, and not the magnetic bearings of the 
lines. Every person will at once perceive that 
much uncertainty in retracing lines will hereafter 
be removed. Had surveys heretofore been made 
according to the true bearings, the surveyor would, 
at the present day, have merely to set the vernier 
or nonius of his compass to the present variation ; 
then the needle would point out on the face of the 
instrument the true courses of the lines of survey, 
by setting it to the proper degree. 

All that surveyors would then have to do, would 
be to go upon the premises to a known corner, and 
run out the true bearings as given in the title, the 
bearings shown on the face of the instrument cor- 
responding thereto. 



102 A MANUAL FOR SURVEYORS. 

We may deduce the true bearing of a former 
survey by the following table or accompanying 
curve, if we know the year the survey was made or 
lines run. This table was formed by a comparison 
of the bearings of lines taken at different periods 
of time. Much difficulty is found in ascertaining 
the date of survey, formerly made. 

Of course this table is given only as an approxi- 
mation merely. It will serve for places north or 
south of Philadelphia, and a few miles east or west 
of the meridian of that place. The table and dia- 
gram are sufficiently plain without explanation. 
The application is as follows. Suppose the mag- 
netic bearing of a line run in 1720 to be N. 45° E., 
what is the true bearing ? "We find by the table 
or diagram the variation in 1720 to be 6J° W., 
the true bearing of the line is therefore N. 38J° E. 

A nonius compass being set as above mentioned 
to the present variation, and a course run N. 38J° 
E., will run out the original line. 

Again : suppose a line in 1810 bore N. 50° W., 
what is the true bearing of the line ? The varia- 
tion in 1810 is found to be 2° W., the true bearing 



A MANUAL FOR SURVEYORS. 103 

is N. 52° W. The needle, as in the former case, 
being set to N. 52° W., will run out the original 
line. 

Any person who is at all acquainted with farm 
surveying, will at once perceive the advantage of 
surveys being made according to the true courses, 
and not the magnetic bearings. Ignorant and in- 
experienced persons will of course object, as more 
skill and knowledge will be brought into requisi- 
tion, and therefore their incompetence will be mani- 
fested. 



Years. 


Variation. 


1682, 


8J° west. 


1690, 


• 8|° " 


1700, 


8° to 8J° west 


1710, 


7J° to 7f °. 


1720, 


*i°- 


1730, 


6i° to 6i°. 


1740, 


5f° to 5i°. 


1750, 


4J° to 4}°. 


1760, 


4J° to 3}. 


1770, 


9-1° to 2-5.° 


1780, 


2i°. 



104: A MANUAL FOR SURVEYORS. 



Years. 


Variation. 


1790, 


1 70 
■■■8 * 


1800, 


1 70 


1810, 


2°. 


1820, 


2f°. 


1830, 


3°. 


1840, 


3|° to 3f °. 


1850, 


4f°. 


1852, 


4f°. 


1853, 


4J° west. 



N. B. If the surveyor find the variation at his 
place for any year, the difference between that va- 
riation, and the variation found on the chart or 
diagram, will be a correction which may be applied 
to variations on the chart, to find the variation, 
nearly, at his place for any given time. 



A MANUAL FOR SURVEYORS. 
Tears. variation west. 




105 



m 



a 

o 

a 
o 






WEST VARIATION. 



10 



106 



A MANUAL FOR SURVEYORS. 



TABLE OF NATURAL TANGENTS, 


Radius 1. 




CO 

OB 

2 

fcO 

to 

p 


b- 

© 
rH 


as 

CO 

o 

CM 


CM 

© 
CO 


Os 

as 

CO 

o 


GO 

O 


T-\ 

to 
o 

rH 
CO 


GO 

cm 

CM 


*o 
o 
-«* 

rH 
CO 


GO 

iO 

T-i 

as 


CO 

co 
t~ 

rH 
O 


© 


00 

t-H 

© 


7-1 

T—l 

© 


i— 1 

© 


b- 

rH 

© 


o 

CO 

T~4 

© 


CO 
CO 

T— 1 

© 


co 
co 

© 


as 

co 

T-{ 

© 


CM 
b- 

T-^ 

© 


to 

b- 

rH 
© 


§ 


t-4 
t— 1 
© 


CM 
CM 

T-H 

© 


cm 

© 


GO 
CM 

T—l 

© 


tH 

CO 
t-4 
© 


CO 

T— 1 

© 


CO 
rH 
© 


o 
o 


CM 

rH 
© 


rH 
© 


o 

CO 


o 

Os 

o 

© 


CO 

as 
o 

© 


CO 

as 

o 
© 


as 

as 
© 
© 


CM 

O 
rH 
© 


o 

© 


CO 

o 

^H 
© 


o 

rH 
rH 
© 


CO 
t-\ 
t-4 
© 


CO 

t-4 
© 


V 

O 


r- 1 

CO 

o 
© 


CO 

o 
© 


CO 

o 
© 


o 
b- 

o 

o 


CO 
b- 

O 

© 


co 

o 
© 


CO 

o 
© 


T~4 

GO 
O 

© 


GO 
O 

© 


b- 

GO 
O 

© 


o 


oi 

00 

o 
© 


o 
co 
o 
© 


GO 
CO 

o 
© 


T-H 

-r44 
o 

o 


O 

© 


CO 

o 
© 


as 

o 
© 


CM 

o 
o 
© 


*0 

o 
© 


CO 

© 
o 


V 

o 


o 

o 

© 


CO 

o 
o 

© 


as 

o 
o 
© 


CM 

t-4 

o 


T-\ 

© 


rH 
© 


o 

CM 

o 


CO 
CM 

© 


co 

CM 


a> 

CM 

© 


CO 

.5 


rH 


CM 


CO 


^ 


o 


CO 


b- 


-GO 


a> 


o 

rH 



It will be sufficiently exact for calculations in 
the field, to take the tang, of the degrees and add 
thereto the tang, of the minutes. 

In many calculations the tangents may be used 
instead of the sines, without material error. 



A MANUAL FOR SURVEYORS. 



10T 



VERSED SIXES, Radius 1°. 





0° 




10° 




20° 


1° 


.00015 = 


7 000 


.01837 = 


i 

50 


.06642 


2 


.00061 = 


1 

3000 


.02185 = 


2 
90 


.07282 


3 


.00137 = 


1 

7 00 


.02563 = 


1 
40 


.07950 


4 


.00211 = 


1 
500 


.02970 = 


1 
35 


.08642 


5 


.00381 = 


3 

Too" 


.03407 = 


1 
3 


.09368 


6 


.00548 = 


5 
900 


.03874 = 


3 
SO 


.10121 


7 


.00745 = 


3 


.04369 = 


4 
90 


.10899 


8 


.00973 = 


1 

Too" 


.04894 = 


1 
20 


; .11705 


9 


.01231 = 


l 


.05448 = 


5 
90 


■ .12538 


10 


.01519 = 


1 
7 


.06031 = 


3 
50 


.13397 



Multiply the versed sine of the elevation of the 
hill by the distance of the slope or surface measure, 
which, being deducted from the slope or oblique 
measure, gives the horizontal distance. 

Example. The oblique distance of a hill of 5° 
elevation is 40 perches. What is the horizontal 
measure ? 



108 A MANUAL FOR SURVEYORS. 

Here .00381 x 40 = .1524, this deducted from 
40, gives 39.85 (40 — .15) per. the distance re- 
quired. 

Or, because .0038 = ^f nearly, we have 40 
_ j**J == 40 — .15 = 39.85 per. as before. 

Note. The numbers in this table might have 
been expressed by vulgar fractions as in the fore- 
going example, which would, in the field, be pre- 
ferable to the decimal form, in many instances 
abridging the calculations, and yet be sufficiently 
correct. This the surveyor can readily perform 
for his own use. 

The oblique measure of any line may be reduced 
to horizontal measure by the traverse table, in the 
following manner : under the degrees of elevation 
or depression, and opposite the oblique distance in 
the distance column, a number will be found in the 
lat. column, which is the horizontal distance ; the 
number in the departure column being the vertical 
altitude of the hill or slope. Taking the preceding 
question, we have, under 5° and distance 40, the 
number 39.85 in a lat. column, which is the hori- 
zontal distance as before. 



